1.2. Cramers rule

Equation (1.2.3) equation (1.2.4) adx bcx = de bf

The determinant contains the coecients of x and y, while 1 is obtained from by

replacing its rst column with the constants on the right-hand sides of (1.2.1) and (1.2.2).Similarly, it can be shown thaty=

2,

(1.2.8)

a e .

2 = c f

where

The determinant 2 is obtained from by replacing its second column with the constantson the right-hand sides of (1.2.1) and (1.2.2).Equations (1.2.7) and (1.2.8) taken together are known as Cramers rule for two unknowns.Example 1.2.1. For the electric circuit shown, the equations for the electric currents (i1and i2 in amperes) are given:8i1 3i2 = 2

3i1 + 10i2 = 6.

Solve the equations using Cramers rule for two unknowns.

2V

i1

i2

6V5

Module 1. Matrices

Solution

1=i1 =

===

i2 =

1.2.2

2683

310310

(2)(10) (3)(6)(8)(10) (3)(3)20 + 1880 92amps.71 8 2

426== 3amps83

713 10

Cramers rule for three unknowns

To solvea1 x + b1 y + c1 z = k1 ,

(1.2.9)

a2 x + b2 y + c2 z = k2 ,

(1.2.10)

a3 x + b3 y + c3 z = k3 ,

(1.2.11)

we can rst eliminate z from the rst two equations by multiplying the rst by c2 and thesecond by c1 and then subtracting. We then obtain the equation(a1 c2 a2 c1 )x + (b1 c2 b2 c1 )y = k1 c2 k2 c1 .

(1.2.12)

Then we can eliminate z from equations (1.2.10) and (1.2.11) by multiplying (1.2.10) by c3and (1.2.11) by c2 and then subtracting. We then obtain the equation(a2 c3 a3 c2 )x + (b2 c3 b3 c2 )y = k2 c3 k3 c2